Hi everybody! I'm preparing an exam of "Analysis II" (that's how the subject's called in German), and I have trouble understanding how to find the limit of a multivariable function, especially when it comes to proving the uniform convergence. Here is an example given in the script of my teacher:(adsbygoogle = window.adsbygoogle || []).push({});

##f(x,y) = \frac{xy}{x^2 + y^2}## (1)

He claims that ##\lim\limits_{x \to 0} ( \lim\limits_{y \to 0} f(x,y)) = \lim\limits_{y \to 0} ( \lim\limits_{x \to 0} f(x,y))## but that ##\lim\limits_{(x,y) \to (0,0)} f(x,y)## still doesn't exist without any explanation. Just before he makes the following claims:

Sufficient conditions for ##\lim\limits_{(x,y) \to (0,0)} f(x,y)## not to exist:

(i) The limits ##\lim\limits_{x \to 0} ( \lim\limits_{y \to 0} f(x,y))## and ##\lim\limits_{y \to 0} ( \lim\limits_{x \to 0} f(x,y))## both exist but are unequal.

(ii) There exists sequences ##x_n \to x_0## and ##y_n \to y_0## so that the sequence ##f(x_1, y_1), f(x_2, y_2),...## does not converge.

(iii) The exists sequences ##x_n \to x_0##, ##\bar{x}_n \to x_0##, ##y_n \to y_0## and ##\bar{y}_n \to y_0## so that the sequences ##f(x_1, y_1), f(x_2, y_2),...## and ##f(\bar{x}_1, \bar{y}_1), f(\bar{x}_2, \bar{y}_2),...## converge but have different limits.

One sufficient condition for ##\lim\limits_{(x,y) \to (0,0)} f(x,y)## to exist:

##\lim\limits_{x \to 0} ( \lim\limits_{y \to 0} f(x,y))## exists and the functions ##f(x,\cdot)## converge uniformly towards the function ##g## on the interval ##(0,r)##. The function ##g## is defined by ##\lim\limits_{x \to 0} f(x,y) = g(y)\ \forall y \in (0,r)##.

The definition of uniform convergence is given at a different place as:

##\forall \epsilon > 0\ \exists \delta > 0\ \forall y \in (0,r)\ \forall x \in (0, \delta): |f(x,y) - g(y)| \leq \epsilon##.

Can you help me make sense of this? As I understand it, a uniformly converging sequence ##f_n## is a sequence for which the objects tend towards the same function ##f## at the same speed (independently from ##x##). Here are my thoughts about the example given above (1):

##\lim\limits_{x \to 0} ( \lim\limits_{y \to 0} f(x,y)) = 0 = \lim\limits_{y \to 0} ( \lim\limits_{x \to 0} f(x,y))##

The limits are indeed equal, and ##g(y) = \lim\limits_{x \to 0} f(x,y) = 0\ \forall y \in (0, + \infty)##. Is that correct? Then if I take the definition of uniform convergence, I must show that for all ##\epsilon##, there exists a strictly positive ##\delta## so that for all ##y \in (0,+ \infty)## and for all ##x \in (0, \delta)## holds good: ##|f(x,y) - g(y)| = |f(x,y)| \leq \epsilon##. Yeah well... I've tried showing the opposite using (ii) but I could not find two sequences that converge so that the sequence of functions diverges. Any tips/clarification about the matter?

Thanks a lot in advance for your answer, I appreciate it!

Julien.

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# I Limits of multivariable functions (uniform convergence)

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